Functional Model of Commutative Operator Systems

A functional model for a commutative system of the linear bounded operators {T₁, T₂}, when T₁ is a contraction, is built.

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Datum:	2008
1. Verfasser:	Zolotarev, V.A.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
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spelling	Zolotarev, V.A. 2016-09-29T19:09:06Z 2016-09-29T19:09:06Z 2008 Functional Model of Commutative Operator Systems / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 3. — С. 420-440. — Бібліогр.: 10 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106516 A functional model for a commutative system of the linear bounded operators {T₁, T₂}, when T₁ is a contraction, is built. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Functional Model of Commutative Operator Systems Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Functional Model of Commutative Operator Systems
spellingShingle	Functional Model of Commutative Operator Systems Zolotarev, V.A.
title_short	Functional Model of Commutative Operator Systems
title_full	Functional Model of Commutative Operator Systems
title_fullStr	Functional Model of Commutative Operator Systems
title_full_unstemmed	Functional Model of Commutative Operator Systems
title_sort	functional model of commutative operator systems
author	Zolotarev, V.A.
author_facet	Zolotarev, V.A.
publishDate	2008
language	English
container_title	Журнал математической физики, анализа, геометрии
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
description	A functional model for a commutative system of the linear bounded operators {T₁, T₂}, when T₁ is a contraction, is built.
issn	1812-9471
url	https://nasplib.isofts.kiev.ua/handle/123456789/106516
citation_txt	Functional Model of Commutative Operator Systems / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 3. — С. 420-440. — Бібліогр.: 10 назв. — англ.
work_keys_str_mv	AT zolotarevva functionalmodelofcommutativeoperatorsystems
first_indexed	2025-11-26T21:13:07Z
last_indexed	2025-11-26T21:13:07Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2008, vol. 4, No. 3, pp. 420�440 Functional Model of Commutative Operator Systems V.A. Zolotarev Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv, 61077, Ukraine E-mail:Vladimir.A.Zolotarev@univer.kharkov.ua Received May 18, 2007 A functional model for a commutative system of the linear bounded operators fT1; T2g, when T1 is a contraction, is built. The construction of functional model is based on an analogue with many parameters of the Lax � Phillips scattering scheme for the isometric dilation U(n) of the semigroup with two parameters T (n) = T n1 1 T n2 2 , where n = (n1; n2) 2 Z 2 +. Key words: functional model, commutative operator system. Mathematics Subject Classi�cation 2000: 47A45. As it is well known, one of the most natural ways of constructing the func- tional model of contraction operator T (kTk < 1) is based on the Lax�Phillips scattering scheme [1]. In this work, the functional model of commutative system of the linear bounded operators fT1; T2g, [T1; T2] = 0, when T1 is a contraction, is obtained using isometric extensions and an analogue with many variables of the Lax�Phillips scattering scheme [2�5]. It is shown that the weight matrices functions of model space have the form which is di�erent from a traditional (the B.S. Pavlov model [1]) one and the structure of given weight functions itself is de�ned by external parameters of iso- metric extensions [2] of the operator system fT1; T2g. The functional model lies in the following: the operator T1 is realized by means of operator of multipli- cation by independent variable in a special function space, the second operator T2 represents the operator of multiplication by meromorphic operator function in the same space. It is typical of the constructed model to di�er crucially from the well-known models in the nonselfadjoint case [6, 7]. 1. Isometric Dilations of Commutative Operator System I. Let a commutative system of the linear bounded operators fT1; T2g, [T1; T2] = T1T2�T2T1 = 0, T1 is a contraction, kT1k � 1, be given in the separable Hilbert space H. Following [2, 3, 8], de�ne the commutative unitary extension for the system fT1; T2g. c V.A. Zolotarev, 2008 Functional Model of Commutative Operator Systems De�nition 1. Let E and ~E be the Hilbert spaces. The collection of mappings V1 = � T1 � K � ; V2 = � T2 �N K � : H �E ! H � ~E; + V 1= � T �1 � �� K� � ; + V 2= � T �2 � ~N� �� K� � : H � ~E ! H �E (1:1) is said to be a commutative unitary extension of the commutative operator system T1, T2 in H, [T1; T2] = 0 if there are such operators �, � , N , � and ~�, ~� , ~N , ~� in the Hilbert spaces E and ~E, respectively, where �, � , ~�, ~� are selfadjoint, and the relations: 1) + V 1 V1 = � I 0 0 I � ; V1 + V 1= � I 0 0 I � ; 2) V �2 � I 0 0 ~� � V2 = � I 0 0 � � ; + V �2 � I 0 0 � � + V 2= � I 0 0 ~� � ; 3) T2�� T1�N = ��; T2 � ~N T1 = ~� ; 4) ~N �� N = K�� ~�K; 5) ~NK = KN (1:2) hold. Consider the following class of commutative systems of the linear operators fT1; T2g [3]. De�nition 2. The commutative operator system fT1; T2g belongs to the class C (T1) and is said to be the contracting T1 operator system if: 1)T1is a contraction, kT1k � 1; 2)E = ~D1H � ~D2H; ~E = D1H � D2H; 3) dimT2 ~D1H = dimE; dimD1T2H = dim ~E; 4) the operators D1j ~E ; ~D1T � 2 �� T2 ~D1H ; ~D1 �� E ; T �2D1jD1T2H are boundedly invertible, where Ds = T �s Ts � I; ~Ds = TsT � s � I; s = 1; 2: (1:3) It is easy to show that if fT1; T2g 2 C (T1), then the unitary extension (1) always exists [2, 3]. II. Recall [1, 9, 10] the construction of unitary dilation U for the contraction T1. Let H be the Hilbert space H = D� �H �D+; (1:4) Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 421 V.A. Zolotarev where D� = l2 Z � (E) and D+ = l2 Z+ ( ~E). De�ne the dilation U on the vector functions f = (uk; h; vk) from H (1.4) in the following way: Uf = � uk�1; ~h; ~vk � ; (1:5) where ~h = T1h+ �u�1, ~v0 = h+Ku�1, ~vk = vk�1, k = 1; 2; : : : . The unitary property of U (1.5) in H follows from 1) (1.2). The construction of isometric dilation [3] of the commutative operator system fT1; T2g 2 C (T1) consists in the continuation of incoming D� and outgoing D+ subspaces D� = l2Z � (E); D+ = l2Z+( ~E) (1:6) by the second variable �n2�. Continue the functions un1 of l2 Z � (E) from semiaxis Z� into domain ~Z2 � = Z�� (Z� [ f0g) = � n = (n1; n2) 2 Z 2 : n1 < 0; n2 � 0 (1:7) using the Cauchy problem [2, 3].( ~@2un = � N ~@1 + � � un; n = (n1; n2) 2 ~Z2 � ; unjn2=0 = un1 2 l2 Z � (E) (1:8) where ~@1un = u(n1�1;n2), ~@2un = u(n1;n2�1). As a result, we obtain the Hilbert space D�(N;�) formed by the solutions un (1.8), besides, the norm in D�(N;�) is induced by the norm of initial data kunk = kun1kl2 Z � (E). Similarly, continue the functions vn1 2 l2 Z+ ( ~E) from semiaxis Z+ into domain Z 2 + = Z+� Z+ using the Cauchy problem( ~@2vn = � ~N ~@1 + ~� � vn; n = (n1; n2) 2 Z 2 +; vnjn2=0 = vn1 2 l2 Z+ (E): (1:9) Denote by D+( ~N; ~�) the Hilbert space formed by solutions vn (1.9), besides, kvnk = kvn1kl2 Z+ ( ~E). Unlike the explicit recurrent scheme (1.8) of the layer-to- layer calculation of n2 ! n2 � 1 for un, in this case of constructing vn in Z 2 +, we have an implicit linear equation system for the layer-to-layer calculation of n2 ! n2 + 1 of function vn. Hereinafter, the following lemma [3] plays an important role. Lemma 1.1. Suppose the commutative unitary extension (Vs, + V s) (1.1) is such that Ker� = Ker � = f0g: (1:10) 422 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems Then KerN \ Ker� = f0g given KerK� = f0g, and respectively Ker ~N� \ Ker ~�� = 0 given KerK = f0g. The solvability of the Cauchy problem (1.9) follows from the given lemma [3]. Consider an operator function of the discrete argument ~�� = � I : � = (1; 0); ~� : � = (0; 1): (1:11) And let Ln0 be the nondecreasing broken line in Z2 + that connects points O = (0; 0) and n = (n1; n2) 2 Z 2 +, the linear segments of which are parallel to the axes OX, n2 = 0, and OY , n1 = 0. By fPkg N 0 denote all integer-valued points from Z 2 +, Pk 2 Z 2 + (N = n1 + n2) that lie on Ln0 , beginning with (0; 0) and �nishing with point (n1; n2), that are numbered in nonascending order (of one of the coordinates Pk). De�ne the quadratic form h~�vki 2 Ln 0 = NX k=0 ~�Pk�Pk�1vPk ; vPk � (1:12) on the vector functions vk 2 D+( ~N; ~�) assuming that P�1 = (�1; 0). Similarly, consider the nonincreasing broken line L�1m in ~Z2 � (1.7) that connects points m = (m1;m2) 2 ~Z2 � and (�1; 0), the linear segments of which are parallel to OX and OY . And let fQsg �1 M , M = m1+m2, be the integer-valued points on L�1m , beginning with m = (m1;m2) and �nishing with (�1; 0), that are numbered in nondescending order (of one of the coordinates Qs). In D�(N;�) de�ne the metric h�uki 2 L �1 m = �1X s=M �Qs�Qs�1 uQs ; uQs � ; (1:13) besides QM �QM�1 = (1; 0) and the operator function �� is de�ned similarly to ~�� (1.11). Denote by ~L�1 �n the broken line in ~Z2 � that is obtained from the curve Ln0 in Z 2 +, n 2 Z 2 +, using the shift by �n� � ~L�1 �n = n Qs = (l1; l2) 2 ~Z2 � : (l1 + n1 + 1; l2 + n2) = Pk 2 Ln0 o : (1:14) III. Having the Hilbert space D�(N;�), that is formed by the solutions of the Cauchy problem (1.8), and the space D+( ~N; ~�), that is formed by the solutions of (1.9), de�ne the Hilbert space HN;� = D�(N;�)�H �D+( ~N; ~�); (1:15) in which the norm is de�ned by the norm of the initial space H = D� �H �D+ (1.4). Denote by Ẑ 2 + the subset in Z 2 +, Ẑ 2 + = Z 2 +n(f0g � N) = f(0; 0)g [ (N � Z+); (1:16) that obviously is an additional semigroup. Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 423 V.A. Zolotarev For every n 2 Ẑ 2 + (1.16), de�ne the operator function U(n) that acts on the vectors f = (uk; h; vk) 2 HN;� (1.15) in the following way: U(n)f = f(n) = (uk(n); h(n); vk(n)) ; (1:17) where uk(n) = PD � (N;�)uk�n (PD � (N;�) is an orthoprojector that corresponds with the restriction onD�(N;�)); h(n) = y0, besides yk 2 H, k 2 Z 2 +, is a solution of the Cauchy problem8< : ~@1yk = T1yk +�u~k; ~@2yk = T2yk +�Nu~k; yn = h; k = (k1; k2) 2 Z 2 +; 0 � k1 � n1 � 1; 0 � k2 � n2; (1:18) at the same time ~k = k � n when 0 � k1 � n1 � 1, 0 � k2 � n2, and, �nally, vk(n) = v̂k + vk�n (1:19) and v̂k = Ku~k + yk, where yk is a solution of the Cauchy problem (1.18). In [3] it is shown that the operator function U(n) (1.17) has the semigroup property and is the isometric dilation of the semigroup T (n) = T n1 1 T n2 2 ; n = (n1; n2) 2 Z 2 +: (1:20) IV. Make a similar continuation of subspaces D� and D+ (1.6) from semiaxes Z� and Z+ by the second variable �n2�, corresponding to the dual situation. By D+ � ~N�; ~�� denote the Hilbert space generated by the solutions ~vn of the Cauchy problem( @2~vn = � ~N�@1 + ~�� ~vn; n = (n1; n2) 2 Z 2 +; ~vnjn2=0 = vn1 2 l2 Z+ ( ~E); (1:21) in which the norm is induced by the norm of initial data k~vnk = kvn1kl2 Z+ (E), besides @1~vn = ~v(n1+1;n2), @2~vn = ~v(n1;n2+1). Continue now every function un1 2 l2 Z � (E) into domain ~Z2 � (1.7) using the Cauchy problem� @2~un = (N�@1 +��) ~un; n = (n1; n2) 2 ~Z2 � ; ~unjn2=0 = un1 2 l2 Z � (E): (1:22) As a result, we obtain the Hilbert space D� (N�;��) generated by ~un, solutions of (1.22), besides k~unk = kun1kl2 Z � (E). The existence of the solution of the Cauchy problem (1.22) follows from Lem. 1. 424 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems De�ne the Hilbert space HN�;�� = D� (N�;��)�H �D+ � ~N�; ~�� ; (1:23) in which the metric is induced by the norm of initial space H = D� � H � D+ (1.4). De�ne the operator function + U (n) for n 2 Ẑ 2 + (1.16) in the space HN�;�� (1.23), which acts on ~f = � ~uk; ~h; ~vk � 2 HN�;�� in the following way: + U (n) ~f = ~f(n) = � ~uk(n); ~h(n); ~v(n) � ; (1:24) where ~vk(n) = P D+( ~N�;~��)~vk+n (PD+( ~N�;~��) is an orthoprojector onD+ � ~N�; ~�� ); ~h(n) = ~y(�1;0), besides ~yk (k 2 ~Z2 � ) satis�es the Cauchy problem 8< : @1~yk = T �1 ~yk + �~v~k; @2~yk = T �2 ~yk + � ~N�~v~k; ~y(�n1;�n2) = h; k = (k1; k2) 2 ~Z2 � ; (1:25) besides ~k = k + n and (�n1 � k1 � �1; �n2 � k2 � 0); �nally, ~uk(n) = ûk + ~uk+n; (1:26) and ûk = K�~v~k +��~yk, where ~yk is a solution of system (1.26). It is clear that the semigroup + U (n) (1.24) is the isometric dilation [3] of the semigroup T �(n), where T (n) has the form of (1.20). Note that the dilations U(n) (1.17) and + U (n) (1.24) are unitary linked, i.e., U� (n1; 0) f = + U (n1; 0) f for all f 2 H (1.4) and for all n1 2 Z+, besides U (n1; 0) on H is a unitary semigroup. 2. Scattering Scheme with Many Parameters and Translational Models I. As it is known [1, 9], the construction of translational (as well as functional) model of contraction T and its dilation U (1.5) follows naturally from the scatter- ing scheme and from the properties of the wave operators W� and the scattering operator S. In order to construct the wave operators W� in the case of many parameters it is necessary [4] to continue the vector functions from l2 Z � ~E � and l2 Z (E) from Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 425 V.A. Zolotarev axis Z into domain Z 2. Continue every function un1 2 l2 Z (E) to the function un, where n = (n1; n2) 2 Z 2, using the Cauchy problem( ~@2un = � N ~@1 + � � un; n 2 Z 2; unjn2=0 = un1 2 l2 Z (E); (2:1) besides kunk = kun1kl2 Z (E). Note that this continuation into the lower half-plane (n2 2 Z�), u (n1; n2)! u (n1; n2 � 1), has a recurrent nature and a continuation into the upper half-plane u (n1; n2)! u (n1; n2 + 1) may be carried out in a non- explicit way in the context of suppositions of Lem. 1.1. As a result, we obtain the Hilbert space l2N;�(E) in which the norm is induced by the norm of initial data. De�ne now the shift operator V (p) V (p)un = un�p; (2:2) where un 2 l2N;�(E) for all p 2 Z 2. Obviously, the operator V (p) (2.2) is isometric. Knowing the perturbed U(n) (1.17) and free V (n) (2.2) operator semigroups, de�ne the wave operator W�(n) W�(k) = s� lim n!1 U(n; k)PD � (N;�)V (�n;�k) (2:3) for every �xed k 2 Z+, where PD � (N;�) is the orthoprojector of narrowing onto the component u�n from l2N;�(E) obtained as a result of continuation into ~Z2 � (1.7) from semiaxis Z� using the Cauchy problem (2.1). It is obvious that W�(0) = W�, where the wave operator W� corresponds with the dilation U (1.5) and the shift operator V in l2 Z (E) [6]. Thus, W�(k) (2.3) is a natural continuation of the wave operator W� onto the �k�th horizontal line in Z 2 when k 2 Z+. Denote by L10;k the broken line in Z 2 + consisting of the two linear segments: the �rst one is a vertical segment connecting points O = (0; 0) and (0; k), where k 2 Z+, and the second segment is a horizontal half-line from point (0; k) to (1; k). Similarly, choose the broken line ~L�1 �1;p in ~Z2 � (1.7) that also consists of the two linear segments, the �rst of which is a half-line from (�1;�p) to point (�1;�p), where p 2 Z+, and the second one is a vertical segment from point (�1;�p) to (�1; 0). In the space HN;� (1.15), specify the following quadratic form: hfi2�(p;k) = h�uni 2 ~L�1 �1;p + khk2 + h~�vni 2 L1 0;k ; (2:4) where corresponding � and ~� in (2.4) are understood in the sense of (1.12) and (1.13). Similarly to (2.4), in l2N;�(E) specify the following �-form: huni 2 �(p;k) = �u�n �2 ~L�1 �1;p + �u+n � L1 0;k ; (2:5) 426 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems where u�n are the continuations of functions from l2 Z � (E) from semiaxes Z�, n2 = 0, obtained by using the Cauchy problem (2.1). Theorem 2.1 [4]. The wave operator W�(k) (2.3) mapping l2N;�(E) into the space HN;� (1.15) exists for all k 2 Z+, and it is an isometry hW�(k)uni 2 �(p;k) = huni 2 �(p;k) (2:6) in metrics (2.4), (2.5) for all p 2 Z+. Moreover, the wave operator W�(k) (2.3) meets the conditions 1) U(1; s)W�(k) =W�(k + s)V (1; s); 2) W�(k)PD � (N;�) = PD � (N;�) (2:7) for all k, s 2 Z+, where PD � (N;�) is an orthoprojector onto D�(N;�). II. Continue the vector functions vn1 from l2 Z � ~E � into domain Z 2 using the Cauchy problem8< : ~@2vn = � ~N ~@1 + ~� � vn; n = (n1; n2) 2 Z 2; vnjn2=0 = vn1 2 l2 Z � ~E � : (2:8) Denote the Hilbert space obtained in this way by l ~N;~� � ~E � , besides kvnk = kvn1kl2 Z( ~E) . Similarly to V (p) (2.2), introduce the shift operator ~V (p)vn = vn�p (2:9) for all p 2 Z 2 and all vn 2 l2~N;~� � ~E � . De�ne the wave operator W+(p) from HN;� into space l2~N;~� � ~E � W+(p) = s� lim n!1 ~V (�n;�p)P D+( ~N;~�)U(n; p) (2:10) for all p 2 Z+; where U(n) has the form of (1.17). It is obvious thatW+(0) =W � +; where W+ is the wave operator [1] corresponding to U (1.5) and to shift ~V in l2 Z � ~E � . Theorem 2.2 [4]. For all p 2 Z+, the wave operator W+(p) (2.11) acting from space HN;� into l2~N;~� � ~E � exists and satis�es the relations 1) W+(p)U(1; s) = ~V (1; s)W+(p+ s); 2) W+(p)PD+( ~N;~�) = P D+( ~N;~�) (2:11) Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 427 V.A. Zolotarev for all p, s 2 Z+; where PD+( ~N;~�) is an orthoprojector onto D+ � ~N; ~� � . Knowing the wave operators W�(k) (2.3) and W+(p) (2.10), de�ne the scat- tering operator in a traditional way [1, 4]: S(p; k) =W+(p)W�(k) (2:12) for all p, k 2 Z+. It is obvious that when p = k = 0, we have S(0; 0) = S; where S is the standard scattering operator, S =W � +W�, for the dilation U (1.5) [1]. Theorem 2.3 [4]. The scattering operator S(p; k) (2.13) represents the bounded operator from l2N;�(E) into l2~N;~� � ~E � , besides 1) S(p; k)V (1; q) = ~V (1; q)S(p + q; k � q); 2) S(p; k)P�l 2 N;�(E) � P�l 2 ~N;~� � ~E � (2:13) for all p, k, q 2 Z+, 0 � q � k; where P� is the narrowing orthoprojector onto solutions of the Cauchy problems (2.1) and (2.9) with the initial data on semiaxis Z� when n2 = 0. III. Following [4], consider the nonnegative operator function Wp;k Wp;k = � W+(p)W � +(p) S(p; k) S�(p; k) W � � (k)W�(k) � (2:14) to de�ne the Hilbert space l2 (Wp;k) = � gn = � vn un � : hWp;kgn; gnil2 <1 � ; (2:15) where un 2 l2N;�(E), vn 2 l2N;� � ~E � . Let W 0 p;0 = � ~V (1; p)W+(p)W � +(p) ~V �(1; p) S(0; p) S�(0; p) I � ; V̂ (1; p) = � ~V �(�1;�p) 0 0 V (1; p) � : (2:16) As it follows from [9], the operator Û(1; p)gn = V̂ (1; p)gn (2:17) acts from the Hilbert space l2 � W 0 p;0 � = � gn = � vn un � : W 0 p;0gn; gn � l2 <1 � (2:150) 428 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems into the space l2 (Wp;0) (2.15). Denote by Ĥp the Hilbert space Ĥp = l2 (Wp;0) P+l 2 ~N;~� � ~E � P�l 2 N;�(E) ! ; (2:18) where P� are orthoprojectors onto solutions of the Cauchy problems (2.1), (2.8) with the initial data on Z�. Consider also Ĥ 0 p = l2 � W 0 p;0 � ~V �(�1;�p)P+l 2 ~N;~� � ~E � V (1; p)P�l 2 N;�(E) ! : (2:180) The spaces Ĥp (2.18) and Ĥ 0 p (2.18 0) are isomorphic one to another, besides, as it is easily seen, the operator Rp : Ĥp ! Ĥ 0 p de�ning this isomorphism has the form Rp = P Ĥ0 p � ~V �(1; p) 0 0 V (�1;�p) � P Ĥp ; (2:19) where P Ĥp and P Ĥ0p are orthoprojectors onto Ĥp (2.18) and Ĥ 0 p (2.180), respec- tively. Specify the operators T̂1 and T̂ (1; p) = T̂1T̂ p 2 , p 2 Z+,� T̂1f � n = P Ĥp fn�(1;0); � T̂ (1; p)f � n = P Ĥp V̂ (1; p) (Rpf)n (2:20) for all fn 2 Ĥp (2.18). Note that the operator T̂1 has the same form (2.20) in all spaces Ĥp (2.28). Theorem 2.4 [4]. Consider the simple commutative unitary extension (Vs, + V s) (2.1) corresponding to the commutative operator system fT1; T2g from the class C (T1) (1.3) and let the suppositions of Lem. 1.1 take place, besides dimE = dim ~E <1. Then the isometric dilation U(1; p) (1.17), p 2 Z+, acting in the Hilbert space HN;� (1.15) is unitary equivalent to the operator Û(1; p) (2.17) mapping the space l2 � W 0 p;0 � (2.15 0) into l2 (Wp;0) (2.15). Moreover, the operators T1 and T (1; p) = T1T p 2 (1.21) speci�ed in H are unitary equivalent to the shift operator T̂1 (2.20) and to the operator T̂ (1; p) (2.20). A similar translational model of dilation + U (n) (1.24) and semigroup T �(n) (1.20) is listed in [4]. Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 429 V.A. Zolotarev 3. Functional Models I. In order to construct the functional models of dilations U(n) (1.17) and + U (n) (1.24), it is necessary to realize the Fourier transformation of translational models from Sect. 2. The Fourier transformation F F (uk) = X k2Z uk� k = u(�); uk 2 l2 Z (E); (3:1) speci�es the isomorphism between l2 Z (E) and the Hilbert space L2 T (E) [1, 9]. Realize the Fourier transformation F (3.1) by variable n1 of every vector function un from the space l2N;�(E), n = (n1; n2) 2 Z 2. Then we obtain (see the Cauchy problem (2.1)) the family of functions u (�; n2) speci�ed on every n2-th horizontal line (n2 2 Z), besides the transition from n2 to n2 � 1 is speci�ed by multiplication by the linear pencil of operators u (�; n2 � 1) = (N� + �)u (�; n2) : (3:2) Note that a corresponding continuation into half-plane n2 2 Z+ may be carried out in the context of suppositions of Lem. 1.1 when dimE <1. As a result, we obtain the Hilbert space of functions u (�; n2), for which (3.2) takes place, besides u(�) = u(�; 0) 2 L2 T (E). We denote this space by L2 T (N;�; E). It is obvious that the shift operator V (p) (2.2), as a result of the Fourier transformation F (3.1) in space L2 T (N;�; E), acts by multiplication V (p)u(�) = �p1(N� + �)p2u(�); (3:3) where u(�) = u(�; 0) and p = (p1; p2) 2 Z 2. Similarly, the Fourier transformation F (3.1) of space l2 Z � ~E � leads us to the Hilbert space L2 T � ~E � . The Fourier transformation F by the �rst variable n1 of every function vn = v(n1;n2) from l2~N;~� � ~E � gives us the family of ~E-valued functions v (�; n2), for which v (�; n2 � 1) = � ~N� + ~� � v (�; n2) (3:4) takes place in view of the Cauchy problem (2.8). The obtained space of functions v (�; n2), where v(�) = v(�; 0) 2 L2 T � ~E � , we denote by L2 T � ~N; ~�; ~E � . As in the previous case, the continuation by rule (3.5), when n+2 2 Z+, is possible when the suppositions of Lem. 1.1 are met and dim ~E < 1. The translation operator ~V (p) (2.9) in the Hilbert space L2 T � ~N; ~�; ~E � is realized by multiplication operator ~V (p)v(�) = �p1 � ~N� + ~� �p1 v(�); (3:5) where v(�) = v(�; 0) and p = (p1; p2) 2 Z 2. 430 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems II. The translational invariance (2.13) of the operator S(p; k) (2.11) signi�es that the Fourier image of the scattering operator S(p; k) represents the operator of multiplication by vector function. In particular, FS(0; 0)uk = S(�)u(�), where u(�) = F (uk) (3.1) and S(�) = K+ (�I � T1) �1� is the characteristic function of extension V1 (1.1) of the operator T1. It follows from relation 1) (2.13) for the operator S(p; k) that it is necessary to �nd the Fourier image of operator S(p; 0) (or of S(0; p), in view of 1) (2.13)) for all p 2 Z+. Further, taking into account the translational invariance of operator S(p; 0), it is obvious that it is su�cient to calculate how S(p; 0) acts on the vector function u0k = uÆk;0, where u is an arbitrary vector from E, and Æk;0 is the Kronecker symbol. For simplicity, consider the case p=1, then it follows from (2.3) and from (2.10) that vmn = ~V (�m;�1)P D+( ~N;~�))U(2m; 1)PD�(N;�)V (�m; 0)u0k ! S(1; 0)u0k when m!1, n 2 Z 2. Elementary calculations show that the vector function vmn is given by vm(n1;0) = � :::; 0; Tm�11 �u; :::; T1�u; �u; Ku ; 0; ::: � ; vm(n1;�1)= � :::; 0; Tm�11 T2�u; :::; T1T2�u; T2�u; (K� + �N)u ;KNu; 0; ::: � ; where the frame signi�es the element corresponding to the null index, n1 = 0. After the limit process, when n! 1 and the Fourier transformation is F (3.1), we obtain that the components v (�; n2) are given by v(�; 0) = S(�)u; v(�;�1) = n KN� +K� + �N + (� � T1) �1 T2� o u: Using now 3) (1.2), we obtain that v(�;�1) = S(�)(N� + �)u: (3:6) Taking into account colligation relations 4), 5) (1.2), we can rewrite the equality (3.6) in the following way: v(�;�1) = � ~N� + ~� � S(�)u: (3:7) De�ne the �kth� characteristic function S(�; k) using the formula S(�; k) = S(�)(N� + �)k; k 2 Z+; (3:8) where S(�) = K + (�I � T1)� and S(�; 0) = S(�). Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 431 V.A. Zolotarev Theorem 3.1. Let uk 2 l2 Z (E) and u(�) = F (uk) (3.1). Then the Fourier transformation F applied to the vector function v = S(p; 0)u represents the family of ~E-valued functions v(�;�k), where 0 � k � p, k 2 Z+, such that v(�;�k) = S(�; k)u(�); (3:9) besides the functions S(�; k) are given by (3.8), 0 � k � p, where S(�; 0) = S(�) = K + (�I � T1) �1 � is the characteristic function of extension V1 (1.1) corresponding to operator T1. Thus the Fourier transformation F of operator S(p; 0) leads us to the operator of multiplication by characteristic function S(�) of the family of functions u (�; n2) from the space L2 T (N;�; E) when n2 2 Z� [ f0g. III. In order to �nd a Fourier image of the weight function Wp;0 (2.14), it is necessary to calculate the Fourier transformation of the operator W+(p)W � +(p) which is also the operator of multiplication by operator function. It follows from the de�nition (2.10) of the wave operator W+(p) that W (n; p) ! W+(p)W � +(p) when n!1, where W (n; p) = ~V (�n;�p)P D+( ~N;~�)U(n; p)U �(n; p)P D+( ~N;~�) ~V �(�n;�p): (3:10) Using the unitary properties of U(n; 0) and ~V (n; 0), n 2 Z, it is easy to ascertain that W (n+ 1; p) = ~V (�n; 0)W (1; p) ~V (n; 0): (3:11) Therefore, it is su�cient to calculate how the operator W (1; p) acts. For simpli- city, conduct calculations for the case of p = 2. Let f = (uk; h; vk) 2 HN;� (1.15) then, using the form of U (1.17), it is easy to show that ~V (�1;�2)P D+( ~N;~�)U(1; 2)f = v̂k � P+vk; (3:12) where P+, as usually, is the orthoprojector in l2~N;~� � ~E � on the subspace of solu- tions of the Cauchy problem (1.9) with the initial data on semiaxis Z+, and the vector function v̂k from ~E is de�ned at points (�1; 0), (�1;�1), (�1;�2) in the following way: v̂�1;0 = h+Ku1;0; v̂�1;�1 = (T2h+�N�1;0) +Ku�1;�1; v̂�1;�2 = fT2 (T2h+�N�1;0) + �Nu�1;�1g+Ku�1;�2: (3:13) Make use of the fact that the function uk is a solution of the Cauchy problem (1.8). Then, taking into account relations 3)�5) (1.2), we obtain that it is possible to write down the relations for the components (3.13), where k = 0, �1, �2, in the following form: 2 4 v̂�1;0 v̂�1;�1 v̂�1;�2 3 5 = 432 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems = 2 4 I 0 0 ~� ~N 0 ~�2 ~N ~� + ~� ~N ~N2 3 5 2 4 h+Ku�1;0 T1h+ �u�1;0 +Ku�2;0 T 2 1 h+ T1�u�1;0 + �u�2;0 +Ku�3;0 3 5 : (3:14) Note that the right-hand member of equality (3.14) is expressed in the terms of operator T1 and external parameters of extension (1.1), and, moreover, the coe�- cients before u�1;k, k = 0, �1, �2, coincide with the corresponding coe�cients of the Laurent factorization of characteristic function S(�) = K + (�I � T1) �1� (1.7) of the operator T1. Introduce into examination the matrices ~L2 = 2 4 I 0 0 ~� ~N 0 ~�2 ~� ~N + ~N ~� ~N2 3 5 ; Q2 = 2 4 0 0 T1 0 0 T 2 1 0 0 3 5 ; R2 = 2 4 K 0 0 � K 0 T1� � K 3 5 : (3:15) Then it follows from (3.14) that the operator W (1; 2) (3.10) is given by W (1; 2) = P�1 ~L2 fQ2Q � 2 +R2R � 2g ~L�2P�1 � P D+( ~N;~�); (3:16) where P�1 is the orthoprojector of narrowing on the vertical line n1 = �1 of grid Z 2 or the operator of multiplication by the Kronecker symbol Æn1;�1. If one makes use of the relations � +KK� = I, T �1 +K� = 0 and T1T � 1 + �� = I that follow from condition 1) (1.2), then it is easy to show that Q2Q � 2 +R2R � 2 = I: (3:17) Therefore, we �nally obtain that W (1; 2) = P�1 ~L2 ~L�2P�1 � P D+( ~N;~�): (3:18) IV. In order to �nd the Fourier transformation of operator W (1; 2) (3.18), calculate the Fourier image of matrix ~L2 (3.15). Let v(�) = v(�; 0) = �1X �1 �kvk 2 L2 T � ~E � , further construct the family of functions v (�; n2) from space L2 T � ~N; ~�; ~E � by rule (3.5) v(�;�k) = � ~N� + ~� �k v(�); k = 0; 1; 2: (3:19) It is easy to make sure that the coe�cients before � in the family of functions v(�;�k) (3.19), where k = 0, 1, 2, correspondingly are equal to v�1;0, ~Nv�2;0 + Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 433 V.A. Zolotarev ~�v�1;0, ~N2v�3;0 + � ~N ~� + ~� ~N � v�2;0 + ~�2v�1;0, which signi�es the application of matrix ~L2 (3.15) to the vector column created by elements v�1;0, v�2;0, v�3;0. Therefore the Fourier transformation F (3.1) of the operator P�1 ~L2 ~L�2P�1 is given by P�1 2 64 I 0 0 ~N� + ~� 0 0� ~N� + ~� �2 0 0 3 75 � 2 64 I ~N�� + ~�� ~N�� + ~�� 2 0 0 0 0 0 0 3 75P�1 2 4 v(�) v(�;�1) v(�;�2) 3 5 ; (3:20) where P�1 is the operator of projection on the subspace � �v , v 2 ~E, and the functions v(�;�k) are constructed by rule (3.19), k = 0, 1, 2. Taking into account the projector P�1, after elementary calculations it is easy to see that the relation (3.20) is equal to ~L2 ~L�2P�1 2 4 v(�) v(�;�1) v(�;�2) 3 5 =W2P�1 2 4 v(�) v(�;�1) v(�;�2) 3 5 : Thus, it follows from (3.10), (3.11) and (3.18) that the Fourier transformation of the operator W+(2)W � +(2) is given by F � W+(2)W � +(2)vn � = fI � P� (I �W2)P�g v (�; n2) ; (3:21) where vn 2 l2~N;~� � ~E � , v (�; n2) 2 L2 T � ~N; ~�; ~E � , W2 = ~L2 ~L�2, and P� is the ortho- projector on the subspace of functions of the type �1X �1 �kvk, vk 2 ~E. To formulate the overall result for all p 2 Z+, de�ne the constant matrix Wp = P0 2 6664 I 0 � � � 0 ~N� + ~� 0 � � � 0 � � � � � � � � � � � �� ~N� + ~� �p 0 � � � 0 3 7775 2 6664 I ~N�� + ~�� ~N�� + ~� �p 0 0 � � � 0 � � � � � � � � � � � � 0 0 � � � 0 3 7775 ; (3:22) where P0 is the operator of narrowing of every component of multiplication (3.22) of matrix (p+ 1)� (p+ 1) on the elements corresponding �0. Theorem 3.2. The Fourier transformation F (3.1) of the operator W+(p)W � +(p), where the operator W+(p) is given by (2.10), is the multiplication 434 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems by constant matrix, F � W+(p)W � +(p)vn � = fI � P� (I �Wp)P�g v (�; n2) ; (3:23) besides Wp is given by (3.22), v (�; n2) = F (vn) 2 L2 T � ~N; ~�; ~E � , where vn 2 l2~N;~� � ~E � and P� is the orthoprojector in L2 T � ~N; ~�; ~E � on the subspace of func- tions v (�; n2) such that v(�; 0) is factorized into the series by powers � �k k2Z � , besides v (�; n2) are obtained from v(�; 0) by rule (3.5). V. It follows from Ths. 3.1 and 3.2 that the operator weight Wp;0 (2.14) after the Fourier transformation F (3.1) is the operator of multiplication by function W (p; �) = � I � P� (I �Wp)P� S(�) S�(�) I � ; (3:24) where Wp is a constant matrix of the type (3.22), and S(�) is the characteristic function of extension V1. After this, it is obvious that the space l 2 (Wp;0) (2.15), as a result of the Fourier transformation F (3.1), is given by L2 T (W (p; �)) = 8< :g(�) = � v(�) u(�) � : 2�Z 0 hW (p; �)g(�); g(�)i d� 2�i� <1 9= ; ; (3:25) where u(�) = u(�; 0) 2 L2 T (E), and is continued to the family of functions u (�; n2) from L2 T (N;F;E) by rule (3.2), and v(�) = v(�; 0) 2 L � ~E � and it also has a continuation to the family v (�; n2) from L2 T � ~N; ~�; ~E � by formula (3.5). Using again Ths. 3.1 and 3.2, it is easy to ascertain that the Fourier image of operator W 0 p;0 (2.15 0) is the operator of multiplication by function W 0(p; �) = " � ~N� + ~� �p fI � P� (I �Wp)P�g � ~N�� + ~�� p S(�) S�(�) I # : (3:26) Therefore, the space l2 � W 0 p;0 � (2.150), after the Fourier transformation F (3.1), is given by L2 T (W 0(p; �)) = 8< :g(�) = � v(�) u(�) � : 2�Z 0 W 0(p; �)g(�); g(�) � d� 2�i� <1 9= ; ; (3:250) where u(�) and v(�) have the same sense as in the de�nition of space L2 T (W (p; �)) (3.25). Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 435 V.A. Zolotarev In view of (3.3) and (3.5), it follows from (2.17) that the dilations U(1; 0) and U(1; p) are the multiplication operators ~U(1; 0)g(�) = �g(�); ~U(1; p)g(�) = � " � ~N�� + ~�� p 0 0 (N� + �)p # g(�); (3:27) where p 2 Z+ and g(�) 2 L2 T (W 0(p; �)). It is easy to see that the model space Ĥp (2.18) after the Fourier transformation is equal to ~Hp = L2 T (W (p; �)) H2 + � ~N; ~�; ~E � H2 � (N;�; E) ! ; (3:28) where the Hardy subspaces H2 � (N;�; E) and H2 + � ~N; ~�; ~E � are obtained from ordinary Hardy classes H2 � (E) and H2 + � ~E � corresponding to domains D � = fz 2 C : jzj > 1g and D + = fz 2 C : jzj < 1g using the rules (3.2) and (3.5), respectively. O b s e r v a t i o n 1. Note that the Hardy space H2 � (N;�; E) contains the functions that are not holomorphic in D � . Really, every function u (�;�n2) = (N� +�)n2u(�), where u(�) 2 H2 � (E) and n2 2 Z+, is factorized into the Fourier series by powers � �k when k 2 (Z�+ n2 � 1). Similarly, the space Ĥ 0 p (2.18 0) after the Fourier transformation F (3.1) is given by ~H 0 p = L2 T � W 0(p; �) � � � ~N�� + ~�� H2 + � ~N; ~�; ~E � �(N� + �)pH2 � (N;�; E) ! ; (3:280) where the weight W 0(p; �) is given by formula (3.26). The isomorphism ~Rp : ~Hp ! ~H 0 p after the Fourier transformation of the operator Rp (2.19) represents ~Rp = P ~H0 p " � � ~N�� + ~�� p 0 0 �(N� +�)�p # P ~Hp ; (3:29) where P ~Hp and P ~H0 p are the orthoprojectors on ~Hp (3.28) and ~H 0 p (3.28 0), respec- tively. Finally, the operators T1 and T (1; p) = T1T p 2 in space (3.28), in view of (3.27), act in the following way:� ~T1f � (�) = P ~Hp �f(�); � ~T (1; p)f � (�) = P ~H� " � ~N�� + ~�� p 0 0 (N� + �)p #� ~Rpf � (�); (3:30) 436 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems where f(�) 2 ~Hp (3.28), and P ~Hp is the orthoprojector on ~Hp (3.28), besides ~Rp is given by (3.29). From this it follows immediately that the initial operator system fT1; T2g, given in H in space ~H1 (3.28), will represent� ~T1f � (�) = P ~H1 �f(�); � ~T2f � (�) = P ~H1 " � ~N�� + ~�� 1 0 0 N� + � #� ~R1f � (�); (3:31) where f(�) 2 ~H1 (3.28). Theorem 3. Consider the simple [2, 3] commutative unitary extension (Vs, + Vs) (1.1) corresponding to the commutative operator system fT1; T2g from the class C (T1) (1.3), and let the suppositions of Lem. 1.1 be met, besides dimE = dim ~E < 1. Then the isometric dilation U(1; p) (1.17) acting in the Hilbert space HN;� (1.15) is unitary equivalent to the functional model ~U(1; 0) (3.27), when p = 0, in L2 T (W 0(p; �)) (3.25 0) and to the operator ~U(1; p) (3.27), when p 2 N, mapping the space L2 T (W 0(p; �)) (3.25 0) into the space L2 T (W (p; �)) (3.25). Moreover, the operators T1 and T (1; p) = T1T p 2 (1.20) given in H are unitary equivalent to the functional model ~T1 (3.30) in ~Hp for all p 2 Z+ and to the operator ~T1(1; p) (3.30) in the concrete model space ~Hp (3.28) when p 2 N. VI. We now turn to the dual situation corresponding to the dilation + U (n) (1.24) in HN�;�� . We list the main results concerning this case without proving. De�ne the constant matrix ~Wp for all p 2 Z+ ~Wp = P0 2 664 I 0 � � � 0 N�� +�� 0 � � � 0 � � � � � � � � � � � �� N�� + �� p 0 � � � 0 3 775 2 664 I N� +� � � � (N� + �)p 0 0 � � � 0 � � � � � � � � � � � � 0 0 � � � 0 3 775 ; (3:32) where P0 is the operator of narrowing on the components corresponding to �0. Consider the weight operator function ~W (p; �) = " I S(�) S�(�) I � P+ � I � ~Wp � P+ # ; (3:33) where the constant matrix ~Wp is given by (3.32). Specify the Hilbert space L2 T � ~W (p; �) � = 8< :g(�) = � v(�) u(�) � : 2�Z 0 D ~W (p; �)g(�); g(�) E d� 2�i� <1 9= ; ; (3:34) where u(�) 2 L2 T (E), v(�) 2 L2 T � ~E � . Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 437 V.A. Zolotarev Moreover, similarly to (3.26), de�ne the weight ~W 0(p; �) = " I S(�) S�(�) (N� + �)�p n I � P+ � I � ~Wp � P+ o (N� + �)p # (3:35) specifying the Hilbert space L2 T � ~W 0(p; �) � = 8< :g(�) = � v(�) u(�) � : 2�Z 0 D ~W 0(p; �)g(�); g(�) E d� 2�i� <1 9= ; ; (3:340) where u(�) and v(�) have the same sense as in the de�nition of space L2 T � ~W (p; �) � (3.34). Specify now the operator functions ~U+(1; 0)g(�) = �g(�); ~U+(1; p)g(�) = � " � ~N� + ~� � �p 0 0 (N� +�)�p # g(�); (3:36) where p 2 Z+ and g(�) 2 L2 T � ~W 0(p; �) � . In this case the model space Ĥp;+ is given by ~Hp;+ = L2 T � ~W (p; �) � H2 + � ~N�; ~��; ~E � H2 � (N�;��; E) ! ; (3:37) where the Hardy spaces H2 � (N�;��; E) and H2 + � ~N�; ~��; ~E � are obtained from the standard Hardy classes H2 � (E) and H2 + � ~E � just as in Subsect. V. Similarly, consider the space ~H 0 p;+ = L2 T � ~W 0(p; �) � � � ~N� + ~� � �p H2 + � ~N�; ~��; ~E � �(N� + �)�pH2 � (N�;��; E) ! ; (3:370) besides the weight ~W 0(p; �) is given by (3.35). Specify the operator ~Rp;+ = P ~H0 p;+ " � � ~N�� + ~�� p 0 0 �(N� + �)p # P ~Hp;+ ; (3:38) where P ~Hp;+ and P ~H0 p;+ are the corresponding orthoprojectors on ~Hp;+ (3.37) and ~H 0 p;+ (3.370). It is clear that the operators T �1 and T �(1; p) = T �1 T �p 2 in space ~Hp;+ are given by � ~T �1 f � (�) = P ~Hp;+ �f(�); 438 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Functional Model of Commutative Operator Systems � ~T �(1; p)f � (�) = P ~Hp;+ � " � ~N� + ~�� p 0 0 (N� + �)�p #� ~Rp;+f � (�) (3:39) for all f(�) 2 ~Hp;+, where P ~Hp;+ is the orthoprojector on ~Hp;+, and ~Rp;+ is given by (3.38). From this it easily follows that the initial operator system fT �1 ; T � 2 g, de�ned in H, in space ~H1;+ (3.37) is given by� ~T �1 f � (�) = P ~H1;+ �f(�); � ~T �2 f � (�) = P ~H1;+ � ~N�� + ~�� 0 0 (N� + �)�1 �� ~R1;+f � (�); (3:40) where f(�) 2 ~H1;+. Theorem 4. Let Vs, + Vs (3.1) be the simple [2, 3] commutative unitary extensions of a commutative operator system fT1; T2g from the class C (T1) (1.3), besides the suppositions of Lem. 1.1 are met and dimE = dim ~E < 1. Then the isometric dilation + U (1; p) (1.24), given in the Hilbert space HN�;�� (3.22), is unitary equivalent to the functional model: ~U+(1; 0) (3.36), when p = 0 in L2 T � ~W (p; �) � (3.34), and to the operator ~U+(1; p) (3.36) mapping the space L2 T � ~W 0(p; �) � (3.34 ) in L2 T � ~W (p; �) � (3.34). Moreover, the operators T �1 and T �(1; p) = T �1 T �p 2 (1.20) given in H are unitary equivalent to the functional model ~T �1 (3.40) in ~Hp;+ (3.37) for all p 2 Z+ and to the operator ~T �1 (1; p) (3.39) only in the concrete model space ~Hp;+ (3.37) when p 2 N. References [1] V.A. Zolotarev, Isometric Expansions of Commutative Systems of Linear Operators. � Mat. �z., analiz, geom. 11 (2004), 282�301. (Russian) [2] V.A. Zolotarev, Analitic Methods of Spectral Representations of Nonselfadjoint and Nonunitary Operators.. MagPress, Kharkov, 2003. (Russian) [3] V.A. Zolotarev, On Isometric Dilations of Commutative Systems of Linear Opera- tors. � J. Math. Phys., Anal., Geom. 1 (2005), 192�208. [4] V.A. Zolotarev, Scattering Scheme with Many Parameters and Translational Models of Commutative Operator Systems. � J. Math. Phys., Anal., Geom. 3 (2007), 424� 447. [5] J.A. Ball and V. Vinnikov, Hardy Spaces on a Finite Bordered Riemann Surface, Multivariable Operator Model Theory and Fourier Analysis Along a Unimodular Curve. In: Systems, Approximation, Singular Integral Operators, and Related Top- ics. Int. Workshop on Operator Theory and Appl., IWOTA (Alexander A. Borichev and Nicolai K. Nikolskii, Eds.), 2000, 37�56. Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 439 V.A. Zolotarev [6] M.S. Liv�sic, N. Kravitsky, A. Markus, and V. Vinnikov, Theory of Commuting Nonselfadjoint Operators. � Math. Appl., Kluver Acad. Publ. Groups, Dordrecht, 332, 1995. [7] V.A. Zolotarev, Time Cones and Functional Model on the Riemann Surface. � Mat. Sb. 181 (1990), 965�995. (Russian) [8] V.A. Zolotarev, Model Representations of Commutative Systems of Linear Opera- tors. � Funkts. Anal. i yego Prilozen. (1988), No. 22, 66�68. (Russian) [9] B. Sz�okefalvi-Nagy and C. Foia�s, Harmony Analisys of Operators in the Hilbert Space. Mir, Moscow, 1970. (Russian) [10] M.S. Brodskiy, Unitary Operator Colligations and their Characteristic Functions. � Usp. Mat. Nauk 33 (1978), No. 4, 141�168. (Russian) 440 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3

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